THESIS DFT CALCULATIONS FOR CADMIUM TELLURIDE (CdTe) Submitted by Sai Avinash Pochareddy Department of Mechanical Engineering In partial fulfilment of the requirements For the Degree of Master of Science Colorado State University Fort Collins, Colorado Fall 2019 Master’s Committee: Advisor: Sampath Walajabad Co-Advisor: Chris Weinberger James Sites Copyright by Sai Avinash Pochareddy 2019 All Rights Reserved ABSTRACT APPLYING QUANTUM ATK TO PERFORM DFT CALCULATIONS ON CADMIUM TELLURIDE (CdTe) Cadmium Telluride (CdTe) thin film photovoltaics (PV) has demonstrated low Levelized Cost of Energy (LCOE). CdTe technology also counted for half the thin film market in 2013 [3]. CdTe PV has the smallest carbon footprint and the energy payback time (less than one year) is the shortest of any current photovoltaic technology. The modules made of CdTe can also be recycled at the end of their lifetime. The attractiveness of these materials comes from their bandgap value (1.5 eV), which falls within the solar spectrum, thereby enabling the efficient creation of electron-hole pairs (or excitons) by solar photons. This has led to the research that dates back to 1950’s and is currently ongoing in many parts of the world. A simple heterojunction cell design was evolved in which p-type CdTe was matched with n-type Cadmium Sulfide (CdS) and by adding the top and bottom contacts. Today, multiple crystalline layers, of thicknesses ranging from a few nanometers (nm) to tens of micrometers (µm), are added to improve the efficiencies of the CdTe PV cells. The highest cell efficiency recorded to date is over 22%. Different computational tools and methods are used to study these effects, with Quantum ESPRESSO and VASP being used for many years now. QuantumATK, built in 2008 by the company Atomistix and acquired by Synopsys in 2017, is a simulation tool that uses Density Functional Theory (DFT) for atomistic-scale modelling of nanostructures. In this work, QuantumATK was used to predict the structural properties of bulk CdTe. Different exchange-correlation (XC) functionals were used to perform the calculations. Firstly, the ii crystal structure of bulk CdTe was predicted using the tool. Later the properties like lattice parameter, were calculated. In addition to structural properties, the electrical properties were also predicted using different XC functionals. Also, the XC functionals that correct the bandgap obtained from the standard functionals were used to predict the bandgaps and the results were also compared to the experimental values again to see how accurately does QuantumATK predicts the electrical properties of bulk CdTe. The LDA and GGA XC functionals, predicted the band gap for bulk CdTe with error percentages of 57% and 40 % respectively, when compared to the experimental value. The more accurate MGGA predicted the band gap with a 24% error while HSE06 (hybrid functional) predicted within 4% of experimental value. The LDA-1/2 and GGA-1/2 predicted the band gap most accurately within 2% of the compared experimental value. All the different XC functionals predicted the crystal structure correctly and the lattice parameter was within 2.2% of the experimental value. iii ACKNOWLEDGEMENTS I would like to thank Dr. Sampath Walajabad, professor in Mechanical Engineering at Colorado State University, for giving me the opportunity to work on this project. This thesis would not have been possible without the trust, support and help from my teacher and advisor. His expertise and leadership helped me to develop an understanding of this project. I would like to thank my thesis committee members, Dr. Chris Weinberger and Dr. James Sites, for their valuable time, guidance and support in this success of this project and my Master’s degree. I would like to extend my special gratitude to Dr. Umberto Martinez of Synopsys for his guidance in this project. His time and help were instrumental in completing this project. I would also like to extend my gratitude to Synopsys team for their timely webinars which helped me understand various new functionals of QuantumATK simulation tool. I would like to extend my gratitude to Department of Mechanical Engineering for the partial funding which helped me work on this thesis. Special thanks to Anthony Nicholson, Aanad Thiyagarajan and Akash Shah, who are also working on DFT projects, for their continuous inputs, knowledge shared and constant feedback, which helped me to a great extent in completing this project. I would not have made it without their help and support. Last but not the least, I would like to thank my parents for trusting and encouraging me in order to achieve my goals. Nothing would have been possible without the support they gave me throughout my life. iv TABLE OF CONTENTS ABSTRACT …………………………………………………………………………………. ii ACKNOWLEDGMENTS ……………………………………………. ……….........................iii LIST OF TABLES ……………………………………………………………………………..vii LIST OF FIGURES ………………………………………………….......................................viii CHAPTER 1: INTRODUCTION ………………………………………………............1 1.1 Cadmium Telluride (CdTe) Photovoltaics …………………………………………...1 1.2 Density Functional Theory (DFT)…………………………………………………....4 1.2.1 Evolution of DFT over time…......................................................................4 1.2.2 Significance of DFT……………………………………………………...6 1.3 Motivation…………………………………………………………………………..6 1.4 Literature Review…………………………………………………………………...7 1.4.1 Schrödinger Equation………………………………………………………7 1.4.2 Approximations to Schrödinger Equation………………………………...10 1.4.2.1 Clamped Nuclei Approximation........................................................11 1.4.2.2 Independent Electron Approximation……………………………....12 1.4.2.3 Mean-Field Approximation………………………………………...14 1.4.3 Hartree-Fock Equations…………………………………...........................15 1.4.4 Hohenberg and Kohn Theorems…………………………………………..16 1.4.5 Kohn-Sham Equations…………………………………………………….19 1.4.6 Exchange and Correlation Functionals…………………………………....20 1.4.6.1 Local Density Approximation (LDA)…………………………...21 1.4.6.2 Generalized Gradient Approximation (GGA)…………………...23 1.4.6.3 Meta-GGA (MGGA)………………………………………….....24 1.4.6.4 Hybrid Functionals……………………………………………....25 1.4.6.5 DFT-1/2 Correction to Functionals……………………………...26 v 1.4.7 Self-Consistent Field (SCF)……………………………………………... 27 1.4.8 Pseudopotentials…………………………………………..........................30 1.4.8.1 Norm-Conserving Pseudopotentials…………………………….31 1.4.8.2 Ultra-Soft Pseudopotentials……………………………………..31 CHAPTER 2: CALCULATING CRYSTAL STRUCTURE OF CdTe USING QUANTUMATK …………………………………………………………………………………………………..33 2.1 Introduction to QuantumATK…………………………………................................33 2.1.1 Linear Combination of Atomic Orbitals (LCAO) Representation..................................................................................................................34 2.1.2 Plane-Wave (PW) Representation………………………….……………..35 2.2 Methods used for Predicting the Crystal Structure of CdTe……..............................36 2.2.1 Crystal Structure Comparison using LDA………………………………..38 2.2.2 Crystal Structure Comparison using GGA……………….…………….....38 2.3 Methods used for Calculating Lattice Parameter of CdTe………………………….39 2.4 Results and Discussions…………………………………………………………….41 2.4.1 Crystal Structure Predicted……………………………………………….41 2.4.2 Lattice Parameter obtained using Different Exchange-Correlation Functionals……………………………………………………………………...43 CHAPTER 3: CALCLATING BAND STRUCTURE OF CdTe USING QUANTUM ATK………………………………………….………………….................................................49 3.1 Calculating Band Structure in Crystals…………………………………………......49 3.2 Band Structure of CdTe……………………………………………………………..53 3.3 Results and Discussions…………………………………………………………….57 3.3.1 Bandgaps Determined by LDA and GGA………………………………...57 3.3.2 Bandgaps Determined by MGGA and HSE06……………………………61 3.3.3 Bandgaps Determined by LDA-1/2 and GGA-1/2.……….........................64 3.3.4 Summary of Results………………………………………………………67 CHAPTER 4: CONCLUSIONS AND FUTURE WORK……………………………………...69 4.1 Structural Properties of CdTe using QuantumATK………………………………...69 4.2 Electrical Properties of CdTe using QuantumATK………………………………...70 vi 4.3 Future Work………………………………………………………………………...72 REFERENCES…………………………………………………………………………………73 vii LIST OF TABLES Table 2.1. Comparison of calculated lattice parameters using different exchange- correlation functionals with experimental value ………………………………………………………........48 Table 3.1. Bandgap comparison with experimental value for LDA & GGA……………….......61 Table 3.2. Bandgap comparison with experimental value for MGGA & HSE06………………64 Table 3.3. Bandgap comparison with experimental value for LDA-1/2 & GGA-1/2………......67 Table 3.4. Summary of bandgaps calculated……………………………………………………67 Table 4.1. Comparisons of calculated properties of CdTe in this experiment to other experimental data available …………………………………………………………………….71 viii LIST OF FIGURES Figure 1.1. CdTe PV installation in Arizona, USA……………………………………………….1 Figure 1.2. CdTe PV installations around the world……………………………………………...2 Figure 1.3. Conversion efficiencies of best solar cells worldwide for various PV technologies since 1976…………………………………………………………………………………………3 Figure 1.4. Superstrate geometry…………………………………………………………….........3 Figure 1.5. Interfaces of CdTe PV…………………………………………………………...........3 Figure 1.6. Electron density in a real system in a given direction………………………….........22 Figure 1.7. Schematic flow chart of self-consistent solutions for Kohn-Sham equations……….29 Figure 2.1. Primitive cells of possible CdTe